Stochastic homogenization of quasilinear PDEs with a spatial degeneracy. (English) Zbl 1180.35591

Delarue, François; Rhodes, Rémi

Asymptotic Anal. 61, No. 2, 61-90 (2009).

This paper is devoted to the study of the stochastic homogenization for some degenerate quasilinear parabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is ergodic, the authors prove the homogenization property for the quasilinear PDEs, by means of the first-order approximation method. The (nonlinear) limit operator need not to be nondegenerate. Concrete examples are provided.

Reviewer: Elisa Alòs (Barcelona)

MSC:

35R60	PDEs with randomness, stochastic partial differential equations
35K65	Degenerate parabolic equations
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

stochastic homogenization; degenerate quasilinear parabolic PDEs; first-order approximation; ergodic operator